Polarization singularities in 2D and 3D speckle fields

Optical skyrmions in the Bessel profile

Optical skyrmions formed in terms of polarization are topological quasi-particles, and they have garnered much interest in the optical community owing to their unique inhomogeneous polarization structure and simplicity in their experimental realization. These structures belong to the Poincaré beams satisfying the stable topology. We theoretically investigated the non-diffracting and self-healing Poincaré beams based on the superposition of two orthogonal Bessel modes by the longitudinal mode matching technique. These Poincaré beams are topologically protected, and we suggest them as optical skyrmions in the corresponding Stokes vector fields. These optical skyrmions are quasi-skyrmions, and their range of propagation depends on the range of superposed Bessel modes. We have shown longitudinal mode matching of superposed Bessel beams is a necessary condition for the generation of propagation-invariant and non-diffracting skyrmions. The proposed longitudinal mode matching technique facilitates the generation of skyrmions with tunable position and range without any on-axis intensity modulations along the propagation axis. A suitable experimental configuration is suggested to realize variable order skyrmions in Bessel modes. The suggested experimental configuration can produce optical skyrmions even in ultra-short laser pulses with high mode conversion efficacy. This work can provide a new direction for the generation of skyrmions with completely new textures and features with reference to existing skyrmions originating from Laguerre-Gaussian modes.

Optimized Stokes imaging for highly resolved optical speckle fields, Part III: topological analysis of polarimetric state distributions with optimized data representations

In this last article of a three-paper series focusing on Stokes polarimetry of optical speckle fields resolved at the individual speckle grain scale, experimental results are provided on test samples of varying nature and polarization properties, and are analyzed extensively. For this purpose, a review of the classical ways of displaying Stokes polarimetric information is provided. Then, some original alternative graphical representations are introduced that ensure optimal readability and interpretability of the Stokes imaging data in the context of speckle field polarimetry, and it is shown how they can be adapted to various observation scales. Finally, these tools are implemented in order to provide a topological analysis of the distribution of the states of polarization across a speckle pattern, and in the vicinity of polarimetric singularities of the field.

Topological Unidirectional Guided Resonances Emerged from Interband Coupling

Unidirectional guided resonances (UGRs) are optical modes in photonic crystal slabs that radiate toward one side without the need for mirrors on the other. In this Letter, we report a mechanism to realize UGRs by tuning the interband coupling effect originating from up-down symmetry breaking. We theoretically find that UGRs that reside along high-symmetric lines correspond to phase singularities of far-field radiation, depicted by phase winding numbers as a type of topological indices. We investigate the phase dislocation lines in three-dimensional parameter space and elaborate on the interplay between UGRs and non-Hermitian degeneracies accordingly. Our findings reveal the topological nature of UGRs about their generation, evolution, and annihilation in general parameter spaces, thus paving the way to new possibilities of light manipulation.

Topological near fields generated by topological structures

The central idea of metamaterials and metaoptics is that, besides their base materials, the geometry of structures offers a broad extra dimension to explore for exotic functionalities. Here, we discover that the topology of structures fundamentally dictates the topological properties of optical fields and offers a new dimension to exploit for optical functionalities that are irrelevant to specific material constituents or structural geometries. We find that the nontrivial topology of metal structures ensures the birth of polarization singularities (PSs) in the near field with rich morphologies and intriguing spatial evolutions including merging, bifurcation, and topological transition. By mapping the PSs to non-Hermitian exceptional points and using homotopy theory, we extract the core invariant that governs the topological classification of the PSs and the conservation law that regulates their spatial evolutions. The results bridge singular optics, topological photonics, and non-Hermitian physics, with potential applications in chiral sensing, chiral quantum optics, and beyond photonics in other wave systems.

Investigation of partially coherent vector vortex beams with non-isotropic states of spatial correlation

In this work, the far-field properties of non-isotropic partially coherent vector vortex beams (PCVVBs) are investigated both theoretically and experimentally. The term non-isotropic signifies that the spatial correlations between the parallel and orthogonal electric field components are distinguishable. It is found that self-orientation and shaping of intensity profile, correlation-induced polarization and depolarization are highly dependent on both the non-isotropic correlation parameters and Poincaré-Hopf index (PHI) of the beam. The simultaneous depolarization and polarization effects are due to the difference in the input correlation parameters that alter the state of polarization (SOP) and degree of polarization (DOP) distributions. The experimental results are in good agreement with the theoretical predictions. The distinguishability of correlation parameters at the source plane leads to significant changes on its intensity profile, DOP, and SOP distributions on far-field propagation, which may found potential applications in beam shaping, detecting and imaging atmospheric lidar, optical imaging and directional transportation where the self-rotation characteristic of beam plays an important role.

Generation of Stokes singularities using polarization lateral shear interferometer

Lateral shear interferometer, being a self-referenced interferometer, has proven to be an important tool in scalar optics. Here we employ a vectorial counterpart - polarization lateral shear interferometer, in which the two interfering beams apart from being derived from the test wavefront, are in orthogonal states of polarization. Therefore when the test wavefront has spatially varying phase gradient across the beam cross-section, the resulting shearogram produces polarization fringes instead of intensity fringes. Further, the shearogram becomes inhomogeneously polarized. This polarization lateral shear interferometer may have potential uses in metrology, but in this article we demonstrate the ability of the interferometer in the generation of all Stokes singularities in the single beam by launching a phase singular beam into it. It is found that a vortex dipole is formed along with other generic Stokes singularities. Experimental observations support the results and they are discussed in the article.

Observing polarization patterns in the collective motion of nanomechanical arrays

In recent years, nanomechanics has evolved into a mature field, and it has now reached a stage which enables the fabrication and study of ever more elaborate devices. This has led to the emergence of arrays of coupled nanomechanical resonators as a promising field of research serving as model systems to study collective dynamical phenomena such as synchronization or topological transport. From a general point of view, the arrays investigated so far can be effectively treated as scalar fields on a lattice. Moving to a scenario where the vector character of the fields becomes important would unlock a whole host of conceptually interesting additional phenomena, including the physics of polarization patterns in wave fields and their associated topology. Here we introduce a new platform, a two-dimensional array of coupled nanomechanical pillar resonators, whose orthogonal vibration directions encode a mechanical polarization degree of freedom. We demonstrate direct optical imaging of the collective dynamics, enabling us to analyze the emerging polarization patterns, follow their evolution with drive frequency, and identify topological polarization singularities.

Coupled nanomechanical resonator arrays serve as model systems to study collective dynamical phenomena. Doster et al. introduce a two-dimensional array of pillar resonators encoding a mechanical polarization degree of freedom for analyzing polarization patterns and identifying topological singularities.

Generation of transversely oriented optical polarization Möbius strips

We report a time-reversal method based on the Richards-Wolf vectorial diffraction theory to generate a prescribed polarization topology on a defined trajectory within areas of relatively high intensity. An example is given to generate transversely oriented optical Möbius strips that wander around an axis perpendicular to the beam propagation direction. A number of sets of dipole antennae are purposefully positioned on a defined trajectory in the y = 0 plane and the radiation fields are collected by one high-NA objective lens. By sending the complex conjugate of the radiation fields in a time-reversed manner, the focal fields are calculated and the optical polarization topology on the trajectory can be tailored to form prescribed Möbius strips. The ability to control optical polarization topologies may find applications in nanofabrication, quantum communication, and light-matter interaction.

Detection of degenerate Stokes index states

Stokes phase is the phase difference between orthogonal component states in the decomposition of any polarization state. Phase singularities in the Stokes phase distribution are Stokes singularities of an inhomogeneous polarization distribution. Under circular decomposition, Stokes phase distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\phi _{12})$$\end{document}(ϕ12) represents polarization azimuth \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\gamma )$$\end{document}(γ) distribution and the singularities present in it are polarization singularities. Therefore, the charge of the Stokes vortices depicted as Stokes index \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{12}$$\end{document}σ12 is an important parameter associated with the polarization singularity. The Hybrid order Poincaré sphere (HyOPS)/Higher order Poincaré sphere (HOPS) beams, all having same Stokes index, contain a Stokes singularity at the center of the beam as these beams are constructed by vortex superposition. These beams, being superposition of orthogonal orbital angular momentum (OAM) states in orthogonal spin angular momentum (SAM) states can offer great multiplexing capabilities in communication. In this article, we identify these degenerate Stokes index states and discuss the ways and means of lifting this degeneracy. Otherwise, there are limitations on intensity based detection techniques, where demultiplexing or segregation of different HOPS/HyOPS beams is warranted. The method adduced here uses the diffraction of these beams through an equilateral triangular aperture in combination with polarization transformation as a probe to lift the Stokes index/Stokes phase degeneracy. Successively, the novelty of the detection scheme is discussed in the context of beams with alike polarization distributions where even the technique of Stokes polarimetry fails to predict the OAM and SAM content of the beam.

Optical framed knots as information carriers

Modern beam shaping techniques have enabled the generation of optical fields displaying a wealth of structural features, which include three-dimensional topologies such as Möbius, ribbon strips and knots. However, unlike simpler types of structured light, the topological properties of these optical fields have hitherto remained more of a fundamental curiosity as opposed to a feature that can be applied in modern technologies. Due to their robustness against external perturbations, topological invariants in physical systems are increasingly being considered as a means to encode information. Hence, structured light with topological properties could potentially be used for such purposes. Here, we introduce the experimental realization of structures known as framed knots within optical polarization fields. We further develop a protocol in which the topological properties of framed knots are used in conjunction with prime factorization to encode information.

Three-dimensional structure of polarization singularities of a light field near a dielectric spherical nanoparticle

We numerically study the structure of polarization singularity lines in a near-field of the sub-wavelength dielectric particle when it is irradiated by a monochromatic elliptically polarized plane wave. For the various values of the ellipticity degree of the incident radiation, we trace the C^T and L^T-lines on which the polarization ellipse turns into a circle or a line, respectively. The continuous variation of the isotropy parameters of the singularities is visualized along the lines. The main rules of C^T and L^T-lines deformation, emergence, and vanishing are revealed.

Generation of polarization singularities with geometric metasurfaces

The polarization singularities are directly generated by using plasmonic metasurfaces with the geometric phase profiles designed to form the Poincaré beams. Different morphologies of polarization topological structures of lemon, star, monstar, spiral, dipole and quadrupole are created by the superpositions of Laguerre–Gauss modes with different orders under orthogonal circular or linear polarization basis. The polarization ellipse patterns and topological features of the produced optical vector fields are analyzed to reveal the properties of the polarization singularities of C-points and L-lines, and the orbital angular momentum states are also measured. The demonstrated polarization singularities generated from the geometric metasurfaces will promise many potential applications related to optical polarization imaging, metrology, optical trapping and quantum information processing.

Geometric phases in 2D and 3D polarized fields: geometrical, dynamical, and topological aspects

Geometric phases are a universal concept that underpins numerous phenomena involving multi-component wave fields. These polarization-dependent phases are inherent in interference effects, spin-orbit interaction phenomena, and topological properties of vector wave fields. Geometric phases have been thoroughly studied in two-component fields, such as two-level quantum systems or paraxial optical waves. However, their description for fields with three or more components, such as generic nonparaxial optical fields routinely used in modern nano-optics, constitutes a nontrivial problem. Here we describe geometric, dynamical, and total phases calculated along a closed spatial contour in a multi-component complex field, with particular emphasis on 2D (paraxial) and 3D (nonparaxial) optical fields. We present several equivalent approaches: (i) an algebraic formalism, universal for any multi-component field; (ii) a dynamical approach using the Coriolis coupling between the spin angular momentum and reference-frame rotations; and (iii) a geometric representation, which unifies the Pancharatnam-Berry phase for the 2D polarization on the Poincaré sphere and the Majorana-sphere representation for the 3D polarized fields. Most importantly, we reveal close connections between geometric phases, angular-momentum properties of the field, and topological properties of polarization singularities in 2D and 3D fields, such as C-points and polarization Möbius strips.

Accurate and rapid measurement of optical vortex links and knots

Optical vortices can evolve in light fields, of which the singularity evolution forms dark lines with complex topological structures, knotted or linked. We propose a method to more accurately and rapidly measure the topology of optical vortex fields. To accurately locate the phase singular points, phase measurement based on digital holography and, further, a numerical search algorithm, are utilized. A motor-driven right-angle prism enables the implementation of a single exposure of hologram for each measurement along the propagation direction, greatly improving the measurement speed. The three-dimensional (3D) spatial distributions of several typical vortex links and knots are experimentally reconstructed. The proposed method is expected to rapidly observe the 3D evolution of other complicated, or even vector, fields.

Simulation of generation and dynamics of polarization singularities with circular Airy beams

The generation and dynamics of polarization singularities have been underresearched for years, while the focusing property of the topological configuration has not been explored much. In this paper, we simulated the generation of low-order polarization singularities with a circular Airy beam and explored the focusing property of the synthetic light field during propagation due to the autofocusing of the component. Our work researched the focusing properties of the polarization singularity configuration, which may help to develop its application prospect.

Topological events on the lines of circular polarization in nonparaxial vector optical fields

In nonparaxial vector optical fields, the following topological events are shown to occur in apparent violation of charge conservation: as one translates the observation plane along a line of circular polarization (a C line), the points on the line (C points) are seen to change not only the signs of their topological charges, but also their handedness, and, at turning points on the line, paired C points with the same topological charge and opposite handedness are seen to nucleate. These counter-intuitive events cannot occur in paraxial fields.

Topological features of vector vortex beams perturbed with uniformly polarized light

Optical singularities manifesting at the center of vector vortex beams are unstable, since their topological charge is higher than the lowest value permitted by Maxwell’s equations. Inspired by conceptually similar phenomena occurring in the polarization pattern characterizing the skylight, we show how perturbations that break the symmetry of radially symmetric vector beams lead to the formation of a pair of fundamental and stable singularities, i.e. points of circular polarization. We prepare a superposition of a radial (or azimuthal) vector beam and a uniformly linearly polarized Gaussian beam; by varying the amplitudes of the two fields, we control the formation of pairs of these singular points and their spatial separation. We complete this study by applying the same analysis to vector vortex beams with higher topological charges, and by investigating the features that arise when increasing the intensity of the Gaussian term. Our results can find application in the context of singularimetry, where weak fields are measured by considering them as perturbations of unstable optical beams.

Numerical generation of a polarization singularity array with modulated amplitude and phase

A point having no defined polarized ellipse azimuthal angle (circularly polarized) in a space-variant vector field is called a polarization singularity, and it has three types: Lemon, Monstar, and Star. Recently, the connection of polarization singularities has been performed. Inspired by this, we conduct a numerical generation of a polarization singularity array. Our method is based on two orthogonal linearly polarized light beams with modulated amplitude and phase. With appropriate distribution functions of amplitudes and phases we can control the polarized states of polarization singularities, which offer a possibility to simulate a polarization singularity array.

Time-reversal constraint limits unidirectional photon emission in slow-light photonic crystals

Photonic crystal waveguides are known to support C-points-point-like polarization singularities with local chirality. Such points can couple with dipole-like emitters to produce highly directional emission, from which spin-photon entanglers can be built. Much is made of the promise of using slow-light modes to enhance this light-matter coupling. Here we explore the transition from travelling to standing waves for two different photonic crystal waveguide designs. We find that time-reversal symmetry and the reciprocal nature of light places constraints on using C-points in the slow-light regime. We observe two distinctly different mechanisms through which this condition is satisfied in the two waveguides. In the waveguide designs, we consider a modest group velocity of vg≈c/10 is found to be the optimum for slow-light coupling to the C-points.This article is part of the themed issue 'Unifying physics and technology in light of Maxwell's equations'.

Simulation of polarized optical speckle fields: effects of the observation scale on polarimetry

In this paper, we propose the simulation of polarized speckle fields using the Stokes formalism, which allows the description of partially polarized electromagnetic waves. We define a unique parameter which determines the partial decorrelation of the involved fields, allowing to simulate the polarized speckles produced by all types of scatterers, from simple to multiple scatterers. We validate this model by comparison with experimental measurements. We use that simulation model to study the impact of the imaging device parameters on polarimetric measurements: first we emphasize a limit of resolution on retardance measurements, then we study the spatial depolarization, which appears when an observer is measuring any space-variant polarization map.

Exceptional polarization structures near the C-lines in diffracted near fields

We study the polarization structures in the vicinity of C-lines in the near fields diffracted from a pair of small holes. We find that, when the incident light is circularly polarized, both the true C-lines and the structures near them are controlled only by the longitudinal component. Furthermore, we find that all the existing singular lines of circular polarization have the winding number ±1, which is very different than the usual numbers ±1/2, and the structure of major axes of the polarization ellipses surrounding these lines are shown to form structures different than the Möbius strip type. All these features prove to be stable upon small changes of shapes or positions of the apertures. However, C-lines with a unit winding number split into two C-lines of half-winding numbers when the incident light is elliptically polarized light.

K-space polarimetry of bullseye plasmon antennas

Surface plasmon resonators can drastically redistribute incident light over different output wave vectors and polarizations. This can lead for instance to sub-diffraction sized nanoapertures in metal films that beam and to nanoparticle antennas that enable efficient conversion of photons between spatial modes, or helicity channels. We present a polarimetric Fourier microscope as a new experimental tool to completely characterize the angle-dependent polarization-resolved scattering of single nanostructures. Polarimetry allows determining the full Stokes parameters from just six Fourier images. The degree of polarization and the polarization ellipse are measured for each scattering direction collected by a high NA objective. We showcase the method on plasmonic bullseye antennas in a metal film, which are known to beam light efficiently. We find rich results for the polarization state of the beamed light, including complete conversion of input polarization from linear to circular and from one helicity to another. In addition to uncovering new physics for plasmonic groove antennas, the described technique projects to have a large impact in nanophotonics, in particular towards the investigation of a broad range of phenomena ranging from photon spin Hall effects, polarization to orbital angular momentum transfer and design of plasmon antennas.

Isotropically polarized speckle patterns

The polarization of the light scattered by an optically dense and random solution of dielectric nanoparticles shows peculiar properties when the scatterers exhibit strong electric and magnetic polarizabilities. While the distribution of the scattering intensity in these systems shows the typical irregular speckle patterns, the helicity of the incident light can be fully conserved when the electric and magnetic polarizabilities of the scatterers are equal. We show that the multiple scattering of helical beams by a random dispersion of "dual" dipolar nanospheres leads to a speckle pattern exhibiting a perfect isotropic constant polarization, a situation that could be useful in coherent control of light as well as in lasing in random media.

Polarized vortices in optical speckle field: observation of rare polarization singularities

Using a recent method able to characterize the polarimetry of a random field with high polarimetric and spatial accuracy even near places of destructive interference, we study polarized optical vortices at a scale below the transverse correlation width of a speckle field. We perform high accuracy polarimetric measurements of known singularities described with an half-integer topological index and we study rare integer index singularities which have, to our knowledge, never been observed in a speckle field.

Polarization singularities and orbital angular momentum sidebands from rotational symmetry broken by the Pockels effect

The law of angular momentum conservation is naturally linked to the rotational symmetry of the involved system. Here we demonstrate theoretically how to break the rotational symmetry of a uniaxial crystal via the electro-optic Pockels effect. By numerical method based on asymptotic expansion, we discover the 3D structure of polarization singularities in terms of C lines and L surfaces embedded in the emerging light. We visualize the controllable dynamics evolution of polarization singularities when undergoing the Pockels effect, which behaves just like the binary fission of a prokaryotic cell, i.e., the splitting of C points and fission of L lines are animated in analogy with the cleavage of nucleus and division of cytoplasm. We reveal the connection of polarization singularity dynamics with the accompanying generation of orbital angular momentum sidebands. It is unexpected that although the total angular momentum of light is not conserved, the total topological index of C points is conserved.

Optical Möbius strips and twisted ribbon cloaks

Optical Möbius strips that surround points of circular polarization, C points, in a generic three-dimensional optical field are cloaked by lines of twisted ribbons attached to the C points. When cloaking occurs, the observable signed twist index that counts the number of half-twists (one or three), and also measures the handedness (right or left), of a generic Möbius strip is determined by the twisted ribbon cloaks. Although some cloaks can be detached, they can never all be removed.

Array of polarization singularities in interference of three waves

In this paper, we show that arrayed polarization singularities (PSs) can be generated by interference of three polarized waves. An experimental setup was proposed and used to produce one of the available types of array structures. We then analytically identified factors that control the structures of these arrays and classified all the structures into three types. Simulation results were used to assist this analysis.

Critical and umbilical points of a non-Gaussian random field

Random fields in nature often have, to a good approximation, Gaussian characteristics. For such fields, the number of maxima and minima are the same. Furthermore, the relative densities of umbilical points, topological defects which can be classified into three types, have certain fixed values. Phenomena described by nonlinear laws can, however, give rise to a non-Gaussian contribution, causing a deviation from these universal values. We consider a random surface, whose height is given by a nonlinear function of a Gaussian field. We find that, as a result of the non-Gaussianity, the density of maxima and minima no longer match and we calculate the relative imbalance between the two. We also calculate the change in the relative density of umbilics. This allows us not only to detect a perturbation, but to determine its size as well. This geometric approach offers an independent way of detecting non-Gaussianity, which even works in cases where the field itself can not be probed directly.

Controlling the polarization singularities of the focused azimuthally polarized beams

We mainly investigate the polarization singularities of the focused azimuthally polarized (AP) beams modulated by spiral phase and sector obstacles. The results reveal that either the spiral phase or sector obstacle can convert the central V-point to C-points, C-point dipoles, or even double V-points under certain conditions. The conversion can be selectively controlled by appropriately setting the topological charge of the spiral phase and the sector angle of the obstacle. These results may have implications for the researches on polarization, focal field manipulation, or even angular momentum of the focused cylindrically polarized beams.

Stochastic geometry and topology of non-Gaussian fields

Gaussian random fields pervade all areas of science. However, it is often the departures from Gaussianity that carry the crucial signature of the nonlinear mechanisms at the heart of diverse phenomena, ranging from structure formation in condensed matter and cosmology to biomedical imaging. The standard test of non-Gaussianity is to measure higher-order correlation functions. In the present work, we take a different route. We show how geometric and topological properties of Gaussian fields, such as the statistics of extrema, are modified by the presence of a non-Gaussian perturbation. The resulting discrepancies give an independent way to detect and quantify non-Gaussianities. In our treatment, we consider both local and nonlocal mechanisms that generate non-Gaussian fields, both statically and dynamically through nonlinear diffusion.

Experimental study on the existence and properties of speckle phase vortices in the diffraction region near random surfaces

We design an optical setup to extract phase vortices in which the interference intensity of reference light wave and speckle fields produced by random screens with different roughness values in the diffraction region near random screens is obtained. Random screens with different roughness are used as samples. Fourier transform is used to extract speckle phase vortices from the interference intensity, and the experimental results show that the phase vortices can be produced when the roughness of the screen is large enough, and they even may appear on the surface. The density of phase vortices would become larger with an increase of the distances in the diffraction region near the random screen. When the distance is certain, the density of phase vortices becomes larger with the increase of roughness. These results would be helpful for understanding the formation of phase vortices.

Poincaré-beam patterns produced by nonseparable superpositions of Laguerre-Gauss and polarization modes of light

We present a study of Poincaré-beam polarization patterns produced by collinear superposition of two Laguerre-Gauss spatial modes in orthogonal polarization eigenstates (circular or linear). We explore theoretically and experimentally the combinations that are possible. We find that the resulting patterns can be explained in terms of mappings of points on the Poincaré sphere onto points in the transverse plane of the beam mode. The modes that we produced yielded many types of polarization singularities.

Optical Möbius strips, twisted ribbons, and the index theorem

The twist numbers of circular optical Möbius strips and twisted ribbons are shown to obey the index theorem under rotation of the plane of observation, and under change in the radius of the path.

Fermionic out-of-plane structure of polarization singularities

A new classification of circular polarization C points in three-dimensional polarization ellipse fields is proposed. The classification type depends on the out-of-plane variation of the polarization ellipse axis, in particular, whether the ellipse axes are in the plane of circular polarization one or three times. A minimal set of parameters for this classification is derived and discussed in the context of the familiar in-plane C point classification into lemon, star, and monstar types. This new geometric classification is related to the Möbius index of polarization singularities recently introduced by Freund.

Topology of light's darkness

We numerically study the topology of optical vortex lines (nodal lines) in volumes of optical speckle, modeled as superpositions of random plane waves. It is known that the vortex lines may be infinitely long, or form closed loops. Loops are occasionally threaded by infinite lines, or linked with other loops. We find the probability of a loop not being threaded decays exponentially with the length of the loop. This behavior has a similarity to scaling laws studied in superfluid turbulence, and polymers modeled as random walks.

Observation of polarization singularities at the nanoscale

With a phase-sensitive near-field microscope we measure independently the two in-plane electric field components of light propagating through a 2D photonic crystal waveguide and the phase difference between them. Consequently, we are able to reconstruct the electric vector field distribution with subwavelength resolution. In the complex field distribution we observe both time-dependent and time-independent polarization singularities and determine the topology of the surrounding electric field.

Polarization singularity anisotropy: determining monstardom

C points, that is, isolated points of circular polarization in transverse fields of varying polarization, are classified morphologically into three distinct types known as lemons, stars, and monstars. These morphologies are interpreted here according to two natural parameters associated with the singularity, namely, the anisotropy of the C point and the polarization azimuth on the anisotropy axis. In addition to providing insight into singularity morphology, this observation applies to the densities of the various morphologies in isotropic random polarization speckle fields.

Generation and dynamics of vortex lattices in coherent exciton-polariton fields

Vortex dynamics in coherent ensembles of exciton polaritons (condensates) is studied in the framework of the polarization-dependent Gross-Pitaevskii equation. Vortex lattices can be resonantly excited in the polariton field by the interference of three or more optical pumps. Vortex-antivortex pairs can also appear in polariton condensates due to scattering with disorder. The nonlinear vortex dynamics is characterized by interactions of vortex cores and vortex-antivortex recombination.